2013A AMC 12 — Official Competition Problems (February 2024)
📅 2024 A 年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 612 人已练习
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1
第 1 题
几何·面积
Square ABCD has side length 10 . Point E is on \overline{BC} , and the area of \bigtriangleup ABE is 40 . What is BE ? [图]
💡 解题思路
We are given that the area of $\triangle ABE$ is $40$ , and that $AB = 10$ .
2
第 2 题
综合
A softball team played ten games, scoring 1,2,3,4,5,6,7,8,9 , and 10 runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? \textbf {(A) } 35 \textbf {(B) } 40 \textbf {(C) } 45 \textbf {(D) } 50 \textbf {(E) } 55
💡 解题思路
To score twice as many runs as their opponent, the softball team must have scored an even number.
3
第 3 题
分数与比例
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
💡 解题思路
We are given that $\frac{6}{10} = \frac{3}{5}$ of the flowers are pink, so we know $\frac{2}{5}$ of the flowers are red.
4
第 4 题
综合
What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\]
💡 解题思路
$\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}$
5
第 5 题
应用题
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid 105 , Dorothy paid 125 , and Sammy paid 175 . In order to share the costs equally, Tom gave Sammy t dollars, and Dorothy gave Sammy d dollars. What is t-d$ ?
💡 解题思路
Simply write down two algebraic equations. We know that Tom gave $t$ dollars and Dorothy gave $d$ dollars. In addition, Tom originally paid $105$ dollars and Dorothy paid $125$ dollars originally. Sin
6
第 6 题
应用题
In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on 20\% of her three-point shots and 30\% of her two-point shots. Shenille attempted 30 shots. How many points did she score?
💡 解题思路
Let the number of 3-point shots attempted be $x$ . Since she attempted 30 shots, the number of 2-point shots attempted must be $30 - x$ .
7
第 7 题
规律与数列
The sequence S_1, S_2, S_3, ·s, S_{10} has the property that every term beginning with the third is the sum of the previous two. That is, \[S_n = S_{n-2} + S_{n-1} for n \ge 3.\] Suppose that S_9 = 110 and S_7 = 42 . What is S_4 ?
💡 解题思路
$S_9 = 110$ , $S_7 = 42$
8
第 8 题
综合
Given that x and y are distinct nonzero real numbers such that x+\tfrac{2}{x} = y + \tfrac{2}{y} , what is xy ?
💡 解题思路
$x+\tfrac{2}{x}= y+\tfrac{2}{y}$
9
第 9 题
几何·面积
In \triangle ABC , AB=AC=28 and BC=20 . Points D,E, and F are on sides \overline{AB} , \overline{BC} , and \overline{AC} , respectively, such that \overline{DE} and \overline{EF} are parallel to \overline{AC} and \overline{AB} , respectively. What is the perimeter of parallelogram ADEF ? [图]
💡 解题思路
Note that because $\overline{DE}$ and $\overline{EF}$ are parallel to the sides of $\triangle ABC$ , the internal triangles $\triangle BDE$ and $\triangle EFC$ are similar to $\triangle ABC$ , and are
10
第 10 题
分数与比例
Let S be the set of positive integers n for which \tfrac{1}{n} has the repeating decimal representation 0.\overline{ab} = 0.ababab·s, with a and b different digits. What is the sum of the elements of S ?
💡 解题思路
Note that $\frac{1}{11} = 0.\overline{09}$ .
11
第 11 题
几何·面积
Triangle ABC is equilateral with AB=1 . Points E and G are on \overline{AC} and points D and F are on \overline{AB} such that both \overline{DE} and \overline{FG} are parallel to \overline{BC} . Furthermore, triangle ADE and trapezoids DFGE and FBCG all have the same perimeter. What is DE+FG ? [图]
💡 解题思路
Let $AD = x$ , and $AG = y$ . We want to find $DE + FG$ , which is nothing but $x+y$ .
12
第 12 题
几何·面积
The angles in a particular triangle are in arithmetic progression, and the side lengths are 4,5,x . The sum of the possible values of x equals a+√(b)+√(c) where a, b , and c are positive integers. What is a+b+c ?
💡 解题思路
Because the angles are in an arithmetic progression, and the angles add up to $180^{\circ}$ , the second largest angle in the triangle must be $60^{\circ}$ . Also, the side opposite of that angle must
13
第 13 题
几何·面积
Let points A = (0,0) , \ B = (1,2), \ C = (3,3), and D = (4,0) . Quadrilateral ABCD is cut into equal area pieces by a line passing through A . This line intersects \overline{CD} at point (\frac{p}{q}, \frac{r}{s} ) , where these fractions are in lowest terms. What is p + q + r + s ?
💡 解题思路
If you have graph paper, use Pick's Theorem to quickly and efficiently find the area of the quadrilateral. If not, just find the area by other methods.
14
第 14 题
规律与数列
The sequence \log_{12}{162} , \log_{12}{x} , \log_{12}{y} , \log_{12}{z} , \log_{12}{1250} is an arithmetic progression. What is x ?
💡 解题思路
Since the sequence is arithmetic,
15
第 15 题
计数
Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?
💡 解题思路
There are two possibilities regarding the parents.
16
第 16 题
统计
A , B , C are three piles of rocks. The mean weight of the rocks in A is 40 pounds, the mean weight of the rocks in B is 50 pounds, the mean weight of the rocks in the combined piles A and B is 43 pounds, and the mean weight of the rocks in the combined piles A and C is 44 pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles B and C ?
💡 解题思路
Let pile $A$ have $A$ rocks, and so on.
17
第 17 题
概率
A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The k^th pirate to take a share takes \frac{k}{12} of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the 12^{th} pirate receive?
💡 解题思路
The first pirate takes $\frac{1}{12}$ of the $x$ coins, leaving $\frac{11}{12} x$ .
18
第 18 题
立体几何
Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2 . The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
💡 解题思路
Set up an isosceles triangle between the center of the 8th sphere and two opposite ends of the hexagon. Then set up another triangle between the point of tangency of the 7th and 8th spheres, and the p
19
第 19 题
几何·面积
In \bigtriangleup ABC , AB = 86 , and AC = 97 . A circle with center A and radius AB intersects \overline{BC} at points B and X . Moreover \overline{BX} and \overline{CX} have integer lengths. What is BC ?
Let S be the set \{1,2,3,...,19\} . For a,b \in S , define a \succ b to mean that either 0 < a - b \le 9 or b - a > 9 . How many ordered triples (x,y,z) of elements of S have the property that x \succ y , y \succ z , and z \succ x ?
💡 解题思路
Imagine that the 19 numbers are just 19 persons sitting evenly around a circle $C$ ; each of them is facing to the center.
21
第 21 题
综合
Consider A = \log (2013 + \log (2012 + \log (2011 + \log (·s + \log (3 + \log 2) ·s )))) . Which of the following intervals contains A ?
💡 解题思路
Let $f(x) = \log(x + f(x-1))$ and $f(2) = \log(2)$ , and from the problem description, $A = f(2013)$
22
第 22 题
概率
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome n is chosen uniformly at random. What is the probability that \frac{n}{11} is also a palindrome?
💡 解题思路
By working backwards, we can multiply 5-digit palindromes $ABCBA$ by $11$ , giving a 6-digit palindrome:
23
第 23 题
几何·面积
ABCD is a square of side length √(3) + 1 . Point P is on \overline{AC} such that AP = √(2) . The square region bounded by ABCD is rotated 90^{\circ} counterclockwise with center P , sweeping out a region whose area is \frac{1}{c} (a π + b) , where a , b , and c are positive integers and gcd(a,b,c) = 1 . What is a + b + c ?
💡 解题思路
We first note that diagonal $\overline{AC}$ is of length $\sqrt{6} + \sqrt{2}$ . It must be that $\overline{AP}$ divides the diagonal into two segments in the ratio $\sqrt{3}$ to $1$ . It is not diffi
24
第 24 题
几何·面积
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?
💡 解题思路
Suppose $p$ is the answer. We calculate $1-p$ .
25
第 25 题
整数运算
Let f : \mathbb{C} \to \mathbb{C} be defined by f(z) = z^2 + iz + 1 . How many complex numbers z are there such that Im(z) > 0 and both the real and the imaginary parts of f(z) are integers with absolute value at most 10 ?
💡 解题思路
Suppose $f(z)=z^2+iz+1=c=a+bi$ . We look for $z$ with $\operatorname{Im}(z)>0$ such that $a,b$ are integers where $|a|, |b|\leq 10$ .